How do I measure the distance between two UV vertices?
I don't believe there is any true way to measure the distance between two vertices in the uv image editor, but there is a fairly simple workaround. 

Finding the (x,y) distance between two points 

With your mesh unwrapped and the UV/image editor open, open the properties panel within the editor via n. At the top there will be what is called "UV vertex". Once you select a vertex it will tell you the coordinates of the vertex.

 Okay, now the hard part. After finding the coordinates of the second vertex use the distance formula, which is:
&radic;&nbsp;(x2−x1)2+(y2−y1)2&nbsp;
 
 This online distance caculator should make things easier. 

Duplicating a UV island 

In 3D View 
First and foremost you must select the desired UV island, like so: 
1. Enter edit mode TAB 
2. Put select mode on "face" CTRL+TAB 
3. Hover mouse over the desired UV island and press L. 
4. Hit Shift+D 
      In UV/Image Editor
Whatever you copy and paste in 3-D view should appear in the image editor. 

References 

How to select a UV island?